2022-05-15

72: Preimage by Product Map Is Product of Preimages by Component Maps

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A description/proof of that preimage by product map is product of preimages by component maps

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About: set

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Starting Context

Target Context


  • The reader will have a description and a proof of the proposition that the preimage by any product map is the product of the preimages by the component maps.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

Main Body

1: Description


For any finite number of sets, S1,i and S2,i where i=1,2,...,k, any corresponding number of maps, fi:S1,i→S2,i, the product map of the maps, fk+1:S1,1×S1,2×...×S1,k→S2,1×S2,2×...×S2,k=(f1,f2,...,fk), and any corresponding number of subsets, S3,i⊆S2,i, the preimage of the product subsets, S3,1×S3,2×...×S3,k by the product map, fk+1−1(S3,1×S3,2×...×S3,k), equals the product of the preimages of the subsets by the component maps, f1−1(S3,1)×f2−1(S3,2)×...×fk−1(S3,k), which is fk+1−1(S3,1×S3,2×...×S3,k)=f1−1(S3,1)×f2−1(S3,2)×...×fk−1(S3,k).

2: Proof


For any element, p=(p1,p2,...,pk)∈fk+1−1(S3,1×S3,2×...×S3,k), fk+1(p)=(f1(p1),f2(p2),...,fk(pk))∈S3,1×S3,2×...×S3,k, so, fi(pi)∈S3,i, so, pi∈fi−1(S3,i), so, p=(p1,p2,...,pk)∈f1−1(S3,1)×f2−1(S3,2)×...×fk−1(S3,k). For any element, p=(p1,p2,...,pk)∈f1−1(S3,1)×f2−1(S3,2)×...×fk−1(S3,k), fk+1(p)=(f1(p1),f2(p2),...,fk(pk)), but fi(pi)∈S3,i, so, fk+1(p)∈S3,1×S3,2×...×S3,k, so, p∈fk+1−1(S3,1×S3,2×...×S3,k).

References
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